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Mirrors > Home > MPE Home > Th. List > distgp | Structured version Visualization version GIF version |
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
distgp.1 | ⊢ 𝐵 = (Base‘𝐺) |
distgp.2 | ⊢ 𝐽 = (TopOpen‘𝐺) |
Ref | Expression |
---|---|
distgp | ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp) | |
2 | simpr 485 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵) | |
3 | distgp.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | fvexi 6677 | . . . . 5 ⊢ 𝐵 ∈ V |
5 | distopon 21533 | . . . . 5 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ 𝒫 𝐵 ∈ (TopOn‘𝐵) |
7 | 2, 6 | syl6eqel 2918 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵)) |
8 | distgp.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
9 | 3, 8 | istps 21470 | . . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵)) |
10 | 7, 9 | sylibr 235 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp) |
11 | eqid 2818 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
12 | 3, 11 | grpsubf 18116 | . . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
14 | 4, 4 | xpex 7465 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V |
15 | 4, 14 | elmap 8424 | . . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) |
16 | 13, 15 | sylibr 235 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵))) |
17 | 2, 2 | oveq12d 7163 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵)) |
18 | txdis 22168 | . . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)) | |
19 | 4, 4, 18 | mp2an 688 | . . . . . 6 ⊢ (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵) |
20 | 17, 19 | syl6eq 2869 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵)) |
21 | 20 | oveq1d 7160 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽)) |
22 | cndis 21827 | . . . . 5 ⊢ (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) | |
23 | 14, 7, 22 | sylancr 587 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
24 | 21, 23 | eqtrd 2853 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) |
25 | 16, 24 | eleqtrrd 2913 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
26 | 8, 11 | istgp2 22627 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
27 | 1, 10, 25, 26 | syl3anbrc 1335 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 𝒫 cpw 4535 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Basecbs 16471 TopOpenctopn 16683 Grpcgrp 18041 -gcsg 18043 TopOnctopon 21446 TopSpctps 21468 Cn ccn 21760 ×t ctx 22096 TopGrpctgp 22607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 df-0g 16703 df-topgen 16705 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cn 21763 df-cnp 21764 df-tx 22098 df-tmd 22608 df-tgp 22609 |
This theorem is referenced by: (None) |
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